Our novel trapped mechanism is more general than the \(Z_{\cal N}\)  framework considered here. It could arise in a large variety of ALP or QCD axion scenarios. For instance, it may apply to axion theories in which an explicit source of PQ breaking is active only at high temperatures and the transition to the true vacuum is non-adiabatic.
Note also that in our scenario kinetic misalignment does not rely on the presence
of non-renormalizable
PQ-breaking operators required in the original formulation [1]}.
It is instead directly seeded by trapped misalignment, which is itself a pure temperature effect.
